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Theorem curry2f 6442
Description: Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
Hypothesis
Ref Expression
curry2.1  |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )
Assertion
Ref Expression
curry2f  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  G : A
--> D )

Proof of Theorem curry2f
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fovrn 6216 . . . . 5  |-  ( ( F : ( A  X.  B ) --> D  /\  x  e.  A  /\  C  e.  B
)  ->  ( x F C )  e.  D
)
213com23 1159 . . . 4  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B  /\  x  e.  A
)  ->  ( x F C )  e.  D
)
323expa 1153 . . 3  |-  ( ( ( F : ( A  X.  B ) --> D  /\  C  e.  B )  /\  x  e.  A )  ->  (
x F C )  e.  D )
4 eqid 2436 . . 3  |-  ( x  e.  A  |->  ( x F C ) )  =  ( x  e.  A  |->  ( x F C ) )
53, 4fmptd 5893 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  ( x  e.  A  |->  ( x F C ) ) : A --> D )
6 ffn 5591 . . . 4  |-  ( F : ( A  X.  B ) --> D  ->  F  Fn  ( A  X.  B ) )
7 curry2.1 . . . . 5  |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )
87curry2 6441 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  C  e.  B )  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
96, 8sylan 458 . . 3  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  G  =  ( x  e.  A  |->  ( x F C ) ) )
109feq1d 5580 . 2  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  ( G : A --> D  <->  ( x  e.  A  |->  ( x F C ) ) : A --> D ) )
115, 10mpbird 224 1  |-  ( ( F : ( A  X.  B ) --> D  /\  C  e.  B
)  ->  G : A
--> D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814    e. cmpt 4266    X. cxp 4876   `'ccnv 4877    |` cres 4880    o. ccom 4882    Fn wfn 5449   -->wf 5450  (class class class)co 6081   1stc1st 6347
This theorem is referenced by:  curry2ima  24097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350
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